Considering then the epoch at which he lived, and the incomplete material which existed in his days for a true science of the heavens, Michael Scot did all that could be reasonably expected of him. He sat at the feet of those who were then the best authorities on this subject. He used his opportunities at Toledo to make the last and most subtle theories of the Moors intelligible to those less fortunate scholars whose attention these must otherwise have escaped. His services to astronomy appeared in the Latin version which he made from a treatise on the _Sphere_ lately composed by Alpetrongi. This author’s name is said to have been, in its Arabic form, Nured-din el Patrugi. Munk, in his _Mélanges_, tells us that the latter designation was derived from a village called Petroches lying a little to the north of Cordova.[144] The Latins corrupted the name in different ways, so that among them it became _Avenalpetrandi_, _Alpetrongi_, or _Alpetragius_. The astronomer who bore it flourished about the year 1190, and is said to have been a renegade, and a scholar of the celebrated Ibn Tofail, the author of the curious Sufic romance called _Hay Ibn Yokhdan_. In the preface to his book on the _Sphere_ Alpetrongi begs to be excused if he has ventured to differ from the tradition of the ancients in his theory of the heavenly movements, and especially from Ptolemy the great master of this science. His apology reminds us that it may be well to examine more exactly than we have yet done the various advances which had been made up to this time by the Arabian astronomy. As early as the ninth century the mathematicians of that nation had simplified the problems of the circle by discovering the way of measurement by sine and tangent instead of by the chord. This improvement is ascribed to Albategni who lived between the years 877 and 929. Calculation was soon made still easier by the invention of algebra. The year 820 is given as the age of Mohammed ben Moussa, surnamed Al Khowaresmi, who had the honour of this important discovery. From the surname of this mathematician the Latins afterwards formed by corruption their common noun _Algorisma_ or _Algorithmus_, from which our word arithmetic is derived.